Angular spheroidal function of the second kind: Series representations (formula 11.09.06.0003)

SpheroidalQS

$Mathematica Notation$

Mathieu and Spheroidal Functions

SpheroidalQS[nu,mu,gamma,z]

Series representations

Generalized power series

Expansions at generic point z==z₀

http://functions.wolfram.com/11.09.06.0003.01

Input Form

SpheroidalQS[\[Nu], \[Mu], \[Gamma], z] \[Proportional] SpheroidalQS[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] + SpheroidalQSPrime[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] (z - Subscript[z, 0]) - (1/(2 (-1 + Subscript[z, 0]^2)^2)) (2 SpheroidalQSPrime[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] Subscript[z, 0] (-1 + Subscript[z, 0]^2) + SpheroidalQS[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] (-\[Mu]^2 - SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] (-1 + Subscript[z, 0]^2) + \[Gamma]^2 (-1 + Subscript[z, 0]^2)^2)) (z - Subscript[z, 0])^2 + (1/(6 (-1 + Subscript[z, 0]^2)^3)) ((-SpheroidalQSPrime[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]]) (-1 + Subscript[z, 0]^2) (-2 + \[Gamma]^2 - \[Mu]^2 - 2 (3 + \[Gamma]^2) Subscript[z, 0]^2 + \[Gamma]^2 Subscript[z, 0]^4 - SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] (-1 + Subscript[z, 0]^2)) + 2 SpheroidalQS[\[Nu], \[Mu], \[Gamma], Subscript[z, 0]] Subscript[z, 0] (-3 \[Mu]^2 - 2 SpheroidalEigenvalue[\[Nu], \[Mu], \[Gamma]] (-1 + Subscript[z, 0]^2) + \[Gamma]^2 (-1 + Subscript[z, 0]^2)^2)) (z - Subscript[z, 0])^3 + \[Ellipsis] /; (z -> Subscript[z, 0])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SpheroidalQS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["SpheroidalQS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", SubscriptBox["z", "0"]]], "]"]], "+", RowBox[List[RowBox[List["SpheroidalQSPrime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", SubscriptBox["z", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]], "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["SpheroidalQSPrime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", SubscriptBox["z", "0"]]], "]"]], " ", SubscriptBox["z", "0"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]]]], "+", RowBox[List[RowBox[List["SpheroidalQS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", SubscriptBox["z", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List[RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[Gamma]", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]], "2"]]]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], "+", RowBox[List[FractionBox["1", RowBox[List["6", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]], "3"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["SpheroidalQSPrime", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", SubscriptBox["z", "0"]]], "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["\[Gamma]", "2"], "-", SuperscriptBox["\[Mu]", "2"], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["3", "+", SuperscriptBox["\[Gamma]", "2"]]], ")"]], " ", SubsuperscriptBox["z", "0", "2"]]], "+", RowBox[List[SuperscriptBox["\[Gamma]", "2"], " ", SubsuperscriptBox["z", "0", "4"]]], "-", RowBox[List[RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["SpheroidalQS", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]", ",", SubscriptBox["z", "0"]]], "]"]], " ", SubscriptBox["z", "0"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", SuperscriptBox["\[Mu]", "2"]]], "-", RowBox[List["2", " ", RowBox[List["SpheroidalEigenvalue", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "\[Gamma]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[Gamma]", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SubsuperscriptBox["z", "0", "2"]]], ")"]], "2"]]]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "3"]]], " ", "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["z", "0"]]], ")"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <msub> <mi> QS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[StyleBox["QS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]]], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["z", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalQS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ∝ </mo> <mrow> <semantics> <mrow> <msub> <mi> QS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[StyleBox["QS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]]], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["z", "0"], SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalQS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> + </mo> <mrow> <semantics> <mrow> <msup> <msub> <mi> QS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox[SubscriptBox[StyleBox["QS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]]]]], "\[Prime]"], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["z", "0"], SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalQSPrime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <semantics> <mrow> <msup> <msub> <mi> QS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox[SubscriptBox[StyleBox["QS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]]]]], "\[Prime]"], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["z", "0"], SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalQSPrime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mrow> <msub> <mi> QS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[StyleBox["QS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]]], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["z", "0"], SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalQS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <semantics> <mrow> <msub> <mi> λ </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mi> γ </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox["\[Lambda]", RowBox[List[TagBox["\[Nu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], "(", TagBox["\[Gamma]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <semantics> <mrow> <msub> <mi> QS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[StyleBox["QS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]]], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["z", "0"], SpheroidalQS, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalQS[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <semantics> <mrow> <msub> <mi> λ </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mi> γ </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox["\[Lambda]", RowBox[List[TagBox["\[Nu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], "(", TagBox["\[Gamma]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <semantics> <mrow> <msup> <msub> <mi> QS </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mi> γ </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SuperscriptBox[SubscriptBox[StyleBox["QS", "IT"], RowBox[List[TagBox["\[Nu]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]]]]], "\[Prime]"], "(", RowBox[List[TagBox["\[Gamma]", SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[SubscriptBox["z", "0"], SpheroidalQSPrime, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[SpheroidalQSPrime[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <msup> <mi> γ </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> μ </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <semantics> <mrow> <msub> <mi> λ </mi> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mi> γ </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox["\[Lambda]", RowBox[List[TagBox["\[Nu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["\[Mu]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]]]]], "(", TagBox["\[Gamma]", SpheroidalEigenvalue, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[SpheroidalEigenvalue[SlotSequence[1]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> SpheroidalQS </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> SpheroidalQS </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <ci> SpheroidalQSPrime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> SpheroidalQSPrime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> SpheroidalQS </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> SpheroidalQS </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -3 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> SpheroidalQSPrime </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> γ </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> μ </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> SpheroidalEigenvalue </ci> <ci> ν </ci> <ci> μ </ci> <ci> γ </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <ci> … </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02